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On the nontrivial solvability of systems of homogeneous linear equations over $ℤ$ in ZFC

Jan Šaroch — 2020

Commentationes Mathematicae Universitatis Carolinae

Motivated by the paper by H. Herrlich, E. Tachtsis (2017) we investigate in ZFC the following compactness question: for which uncountable cardinals $κ$ , an arbitrary nonempty system $S$ of homogeneous $ℤ$ -linear equations is nontrivially solvable in $ℤ$ provided that each of its subsystems of cardinality less than $κ$ is nontrivially solvable in $ℤ$ ?

Bi-ideal-simple semirings

Václav Flaška; Tomáš Kepka; Jan Šaroch — 2005

Commentationes Mathematicae Universitatis Carolinae

Commutative congruence-simple semirings were studied in [2] and [7] (but see also [1], [3]--[6]). The non-commutative case almost (see [8]) escaped notice so far. Whatever, every congruence-simple semiring is bi-ideal-simple and the aim of this very short note is to collect several pieces of information on these semirings.

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On the nontrivial solvability of systems of homogeneous linear equations over ℤ in ZFC

Bi-ideal-simple semirings

On the nontrivial solvability of systems of homogeneous linear equations over $ℤ$ in ZFC